The certitude of conclusions

Math is the paradigm instance of proof, and one of the things it shows us is that the relation between our initial beliefs and the conclusion we prove can have any measure of certitude. Sometimes math proves a conclusion that is so counterintuitive we have a hard time accepting it even after we see it demonstrated,* sometimes it proves something that seems so self-evident we don’t see the point of the proof at all, sometimes by the time the proof is done we are less certain about the conclusion than when we started.**

Proofs in metaphysics touch on subjects that we often find ourselves deeply committed to in one way or another, and so different persons at different times will relate in different ways to the proof. Sometimes it seems that all we’re trying to do is clarify the basis of a self-evident axiom, other times the conclusion is so off-putting we figure there has to be some problem with it somewhere. When it comes to the big topics of God, nature, freedom, mind, authority, human nature, etc. the same conclusion that is axiomatic at one time seems impossible at another. STA notes that in his own time people viewed demonstrations for the existence of God superfluous, and so for him the proof was mostly a formal requirement of his Aristotelian attempt to build up a science about God, but for us the same proof is a scandal to our views of the universe, which even believers see as a self-sufficient system that God can only be present in by intrusion. Again, when I give Rawls’s veil of ignorance argument to students it strikes them as something they’ve believed forever, but if we gave the same argument to STA he’d find it as bizarre as the thought that you could assign family members to arbitrary roles in the family.

The point is not to argue for historical relativity but to point out a dimension of what it is to prove something. Proofs do not universally take us from an initial hypothesis with 50-50 probability to a final conclusion with a probability of 1. It’s equally silly to think that we haven’t proved something until we’ve counted up the number of persons who we’ve convinced. If all we wanted to do was convince people of some metaphysical premise the smart play is to drop proofs altogether and use marketing, peer pressure, advertising jingles, rituals, opportunities to enhance one’s reproductive fitness by believing the premise, etc.

Proofs are about trying to build up a body of knowledge, either for its own sake or the sake of some action. They are a fantastically ineffective approach to getting people to commit themselves to something, notwithstanding the non-zero percentage of the population who are committed to something because of a proof.

*Many see Euclid 3.16 like this, or Poincare’s proof that there is a possible geometry where a straight-line is perpendicular to itself.

**Russell and Whitehead taking an entire book to prove 1+1=2 is a case in point.